1
WITHDRAWAL OF PAIRS OF THREADED RODS WITH SMALL EDGE DISTANCES AND 1
SPACINGS 2
3
Haris Stamatopoulos1* and Kjell Arne Malo1 4
5
1Department of Structural Engineering, Norwegian University of Science and Technology (NTNU), 6
Rich. Birkelandsvei 1A, 7491, Trondheim, Norway.
7
* Corresponding author, Tel: +47 735 94675, Email: [email protected] 8
9
ABSTRACT: An experimental investigation on withdrawal of pairs of screwed-in threaded rods embedded in 10
glued-laminated timber elements is presented in this paper. Specimens with varying angles between the rod axis 11
and the grain direction (α= 15, 30, 60, 90°) and 2 different configurations with respect to edge distances and 12
spacings were tested. The diameter and the embedment length of the rods were 20 mm and 450 mm, 13
respectively. The threaded rods were embedded in a row perpendicular to the plain of the grain. The edge 14
distances and spacings were smaller than the minimum requirements according to Eurocode 5. The withdrawal 15
capacity of pairs of rods was compared to the withdrawal capacity of single rods and the effective number, nef, 16
was found to be in the range 1.72-1.94, despite the small edge distances and spacings. Based on the obtained 17
experimental results, a simple approximating expression was derived for nef. An analytical model based on 18
Volkersen theory with an idealized bi-linear constitutive relationship was used to estimate the withdrawal 19
capacity and stiffness. The analytical estimations were in good agreement with the experimental results. Finally, 20
the withdrawal stiffness was estimated by use of finite element simulations. The numerical estimations for the 21
withdrawal stiffness were also in good agreement with the experimental results.
22 23
KEYWORDS: Threaded rod, withdrawal, edge distance, spacing, rod-to-grain angle 24
25
1 INTRODUCTION 26
1.1 BACKGROUND 27
A remarkable increased use of axially loaded self-tapping screws and threaded rods, either as reinforcements or 28
as fasteners in timber structures, has taken place in recent years. Threaded rods show in general high withdrawal 29
capacity and stiffness, and thus they may be used to develop strong and stiff connections. In many practical cases 30
multiple axially loaded threaded rods are used. Studies (Blaß and Laskewitz 1999; Gehri 2009; Krenn and 31
Schickhofer 2009; Mahlknecht et al. 2014; Mori et al. 2008) have shown that connections with multiple axially 32
loaded screws or glued-in rods can be very efficient, as each rod can reach a capacity in the order of 80-100% of 33
the capacity of the respective single rod case. These studies have mainly focused on cases where the screws/rods 34
were placed parallel or perpendicular to the grain direction. The background given in this Section is focusing on 35
solid timber and glued-laminated timber (abbr. glulam).
36
The effectiveness of connections with multiple axially-loaded fasteners may be influenced by insufficient edge 37
and end distances or spacings, as failure modes other than withdrawal or steel failure may be triggered and the 38
2
full tensile capacity may not be reached. In order to take this into account, modern design codes and technical 39
approvals set restrictions on the minimum edge and end distances as well as spacings. Typically, the minimum 40
edge and end distances and spacings are provided as multiple of the outer thread diameter d. The minimum edge 41
and end distances and spacings for screws according to EN1995 (abbr. EC5) (CEN 2004) are provided in Table 42
1. The associated definitions are specified in Fig. 1. As shown in Fig. 1, screws may be installed in rows parallel 43
to the plane of the grain and thus sharing the same plane of the grain (for example screws 1-3-5 and 2-4-6) or in 44
rows perpendicular to the plane of the grain and thus placed in different grain planes (for example screws 1-2, 3- 45
4 and 5-6). For shortness, the former configuration is denoted as ‘in-series’ and the latter as ‘in-parallel’, confer 46
Fig. 1.
47 48
Table 1: Minimum edge and end distances and spacings for screws according to EC5 (CEN 2004) 49
a1 7d
a1.CG 10d
a2 5d
a2.CG 4d
50
51
Fig. 1 Definitions of edge and end distances and spacings according to EC5 (CEN 2004) and naming of 52
configurations 53
3
Small spacings (a1, a2) may lead to block/plug shear failures. Mahlknecht et al (2014) have shown that block 54
shear failure may occur even if the minimum requirements given in Table 1 are fulfilled. For screws inserted 55
parallel to the grain, small edge distances may lead to splitting failure (Nakatani and Walford 2010). Insertion of 56
screws parallel to the grain without some sort of reinforcement against splitting should be avoided, because 57
tensile stresses perpendicular to the grain may develop due to other reasons than withdrawal (for example 58
moisture-induced stresses (Angst and Malo 2012)). In order to eliminate the risk of splitting, EC5 (CEN 2004) 59
imposes the minimum edge and end distances of Table 1 and it does not allow for installation of screws in an 60
angle to the grain direction less than 30°. According to DIN 1052:2008-12 (DIN 2008), the insertion of screws in 61
pre-drilled holes has a positive effect in comparison to self-tapping screws and thus the minimum requirements 62
for edge distances and spacings are less strict. However, this positive effect of pre-drilling is not taken into 63
account in EC5 (CEN 2004).
64
According to EC5 (CEN 2004), the withdrawal capacity of connections with multiple axially loaded screws 65
(which comply with the requirements in Table 1) is determined by multiplying the corresponding capacity of a 66
single screw with the effective number of screws, nef, given by:
67
=
0.9n
efn
(1)where n is the number of screws acting together in a connection. The background research for Eq.(1) could not 68
be located by the present authors. According to recent studies (Krenn and Schickhofer 2009; Mahlknecht et al.
69
2014), the effective number of screws is equal to the actual number with respect to withdrawal capacity (i.e. nef= 70
n) for configurations where the requirements of Table 1 are fulfilled (and therefore Eq. (1) is conservative 71
according to these studies).
72
EC5 (CEN 2004) does not provide guidelines for the estimation of withdrawal stiffness of axially loaded self- 73
tapping screws or threaded rods. In the case of single self-tapping screws with diameters up to 12-14 mm, most 74
research effort has been devoted on the determination of the withdrawal strength and the influence of several 75
parameters on the withdrawal strength. Fewer results are available for the withdrawal stiffness. Some simple 76
expressions for the withdrawal stiffness can be found in several technical approvals, see for example Z-9.1-472 77
(DIBt 2011) or ETA-11/0190 (DIBt 2013). However, the expressions found in these technical approvals: a) are 78
different with respect to the effect of the diameter and the embedment length (possibly due to different 79
experimental configurations), b) they do not take into account the influence of the angle between the screw and 80
the grain direction and c) they cannot be extrapolated for threaded rods with greater dimeters (Stamatopoulos 81
2016). A proposal for the withdrawal stiffness of single axially loaded self-tapping screws based on a more 82
systematic approach and a huge sample of experimental results can be found in (Ringhofer et al. 2015). In the 83
case of multiple axially-loaded self-tapping screws the existing results for the withdrawal stiffness are sparse.
84
Krenn and Schickhofer (2009), based on experimental results of axially loaded joints with inclined self-tapping 85
screws and steel plates as outer members, have proposed the following effective number of screws for design in 86
the serviceability limit state:
87 88
4
=
0.8 ef.sern n
(2)To the knowledge of the authors, there are no guidelines available for the withdrawal stiffness of single threaded 89
rods with greater diameters in the present European technical approvals. Experimental results covering both 90
withdrawal capacity and stiffness can be found in (Nakatani and Komatsu 2004) for rods with varying 91
embedment length embedded parallel to the grain and in (Blaß and Krüger 2010) for rods with varying 92
embedment length and diameter embedded with an angle of 45° and 90° to the grain direction. Another 93
investigation on the withdrawal stiffness of single threaded rods with varying embedment lengths (l= 100, 300, 94
450, 600 mm) and rod-to-grain angles (α= 0, 10, 20, 30, 60 and 90°) by use of experimental, numerical and 95
analytical methods has recently been presented by the present authors (Stamatopoulos and Malo 2016). The 96
corresponding results with respect to the withdrawal capacity are given in (Stamatopoulos and Malo 2015a;
97
Stamatopoulos and Malo 2015b).
98
In the case of multiple axially loaded threaded rods, the available experimental results are very sparse. (Mori et 99
al. 2008) presented an experimental study of configurations with 1, 2 and 4 rods with varying spacing embedded 100
parallel and perpendicular to the grain in glulam elements. According to this investigation, the full withdrawal 101
capacity could be reached for spacing equal to 4 times the diameter whereas 80%-90% of the withdrawal 102
capacity of a single rod was reached for specimens with spacing equal to 2 times the diameter. Similar results 103
have been obtained by (Gehri 2009) in an investigation of the influence of spacing on withdrawal strength of 104
self-tapping screws with a diameter of 10 mm embedded parallel to the grain (in this case the threshold spacing 105
to reach the full withdrawal capacity was 5 times the diameter). With respect to withdrawal stiffness, (Mori et al.
106
2008) could not reach definite conclusions with respect to the withdrawal stiffness of multiple threaded rods.
107
108
1.2 OUTLINE 109
For a pair of threaded rods installed ‘in parallel’ in a timber element, the minimum required width is equal to 13d 110
if the requirements of Table 1 are fulfilled. In practice however it may be desirable to install the rods with 111
smaller edge distances and spacings. In the present study, only configurations with a pair of threaded rods 112
installed in ‘parallel’ were investigated. In this configuration, plug shear failure cannot occur because shear 113
stresses are concentrated towards the plane of the grain and the rods are embedded in different planes. On the 114
contrary, in ‘series’ configurations with long, axially-loaded fasteners installed with small spacings are prone to 115
plug shear failure because the fasteners share the same plane of the grain. The difference in the failure mode of 116
the two configurations is illustrated in Fig. 2. Fig. 2a is taken from the present study and Fig. 2b is taken from a 117
study on screws’ withdrawal from laminated veneer lumber elements (Carradine et al. 2009).
118 119
5 120
Fig. 2 Failure modes of configurations with small spacings: (a) ‘in parallel’ configuration, (b) ‘in series’
121
configuration (Carradine et al. 2009) 122
123
For long rods inserted with an inclination to the grain direction a splitting crack may form along the grain if the 124
distance to the edge a2.CG is small. However, failure due to this crack may be prevented because the crack is 125
bridged by the rod itself and any additional reinforcement against splitting which may exist. In the present study, 126
the withdrawal of pairs of screwed-in threaded rods installed in ‘parallel’ with edge distances and spacings 127
which do not comply with the minimum requirements of EC5 (CEN 2004) is investigated. Experimental, 128
analytical and numerical methods are used.
129 130
2 MATERIALS AND METHODS 131
2.1 EXPERIMENTAL 132
The experimental set-up used for the withdrawal tests is presented in Fig. 3. The loading condition of the 133
specimens was a ‘remote’ pull-push (i.e. the support was provided in the same plane surface as the entrance of 134
the rods, but at a distance to the rods). The distance between the supports was s = 185 mm. A thin steel plate, as 135
shown in Fig. 3d, was placed between the supports and the specimen in order to counteract tensile stresses due to 136
bending, while allowing for local deformation on the surface of the specimen. The relative displacement between 137
the rods and the supports was measured by two displacement transducers, attached to a steel apparatus clamped 138
on the rods, as shown in Fig. 3f. The average of these two measurements was used for the displacement. To 139
ensure equal deformation of the rods a very stiff coupling part was used, confer Fig. 3g.
140
6 141
Fig. 3 Experimental set-up: (a) virtual 3D representation, (b) top view, (c) side view, (d) steel plate, (e) 142
configurations A and B, (f) attachment of LVDTs and (g) photo 143
144
The specimens were cut from several glulam beams of Scandinavian class L40c which is a combined type of 145
glulam corresponding to the European strength class GL30c (CEN 2013). This type of glulam is fabricated with 146
45 mm thick lamellas, made of Norwegian spruce (Picea Abies). Its mean and characteristic density are ρm= 470 147
kg/m3 and ρk= 400 kg/m3, respectively. For increased homogeneity, all specimens were manufactured such that 148
the rods were embedded in the inner, weaker lamellas of the beams. Moreover, specimens were cut in such a 149
way so that the rods were embedded in different laminations within the same specimen series. The width, b, of 150
the glulam beams and consequently of all specimens was equal to 140 mm. SFS WB-T-20 (DIBt 2010) steel 151
7
threaded rods were used. The outer-thread diameter, d, of the rods was 20 mm and the core diameter, dc, was 15 152
mm. Prior to screwing-in of rods, all specimens were pre-drilled with a diameter equal to the core diameter of the 153
rods, i.e. 15 mm. All specimens were conditioned to standard temperature and relative humidity conditions 154
(20°C / 65% R.H.), leading to approximately 12% moisture content in the wood.
155
Specimens with two configurations with respect to the edge distances and spacings were tested as shown in Fig.
156
3e. In configuration A the rods were installed with spacing a2= 2d and edge distance a2.CG= 2.5d. In 157
configuration B the rods were installed with spacing a2= 4d and edge distance a2.CG = 1.5d. These specific values 158
were chosen so that the minimum net distance between the rods (in configuration A) or the minimum net 159
distance between the rod and the edge (in configuration B) were equal to the diameter of the rods. The 160
embedment length of the rods in all specimens was l = 450 mm. Specimens with 4 different rod-to-grain angles 161
were tested (α = 15, 30, 60, 90°). Two tests were performed for each configuration and rod-to-grain angle 162
resulting in a total number of 16 tests. The specimens are denoted Sα-configuration-no based on their rod-to- 163
grain angle α, their configuration (A or B) and the serial number of the test. Testing was performed using the 164
loading protocol given in EN 26891:1991 (ISO6891:1983) (CEN 1991). The test program is summarized in 165
Table 2.
166 167
Table 2: Test program 168
Specimen b×h×L ρma α l d / dc a2 a2.CG Series (mm) (kg/m3) (deg) (mm) (mm) (mm) (mm)
S15-A-(1-2) 140×545×1200 481.3 15 450 20 / 15 40
(2d)
50 (2.5d)
S15-B-(1-2) 140×545×1200 481.3 15 450 20 / 15 80
(4d)
30 (1.5d)
S30-A-(1-2) 140×545×940 482.2 30 450 20 / 15 40
(2d)
50 (2.5d)
S30-B-(1-2) 140×545×940 482.6 30 450 20 / 15 80
(4d)
30 (1.5d)
S60-A-(1-2) 140×545×765 469.6 60 450 20 / 15 40
(2d)
50 (2.5d)
S60-B-(1-2) 140×545×765 485.5 60 450 20 / 15 80
(4d)
30 (1.5d)
S90-A-(1-2) 140×545×500 472.6 90 450 20 / 15 40
(2d)
50 (2.5d)
S90-B-(1-2) 140×545×500 476.5 90 450 20 / 15 80
(4d)
30 (1.5d)
a Determined for the whole specimen 169
170
2.2 ANALYTICAL 171
Analytical expressions for withdrawal capacity, stiffness and the stress and displacement distributions of single 172
rods can be found in (Stamatopoulos and Malo 2015b; Stamatopoulos and Malo 2016). This model is based on 173
classical Volkersen theory (Volkersen 1938) applied to axially loaded fasteners (Jensen et al. 2001). It is 174
assumed that all shear deformation, δ(x), occurs in a shear zone of finite dimensions and it is related to the mean 175
interfacial shear stress, τ(x), by a bi-linear constitutive τ(x)- δ(x) relationship. An example of a real non-linear 176
behaviour is compared to the modelled bi-linear relationship in Fig. 4. The bi-linear idealization separates the 177
curve in two distinct domains; the linear elastic domain and the fracture domain. These domains are 178
characterized by the equivalent shear stiffness parameters Γe and Γf, which are the slopes of the two branches of 179
the bi-linear constitutive relationship.
180
8 181
182
Fig. 4 Real and idealized bi-linear τ(x)-δ(x) curve 183
184
The model can be extended to a group of multiple rods symmetrically installed ‘in parallel’ under the assumption 185
that Γe and Γf, are the same for all rods. For the pull-push or the pull-shear loading condition, the withdrawal 186
stiffness, Kw, is equal to:
187
= tanh
nw ef.ser ef e
n
K n π d l Γ ω ω
(3)
The effective number of rods for the serviceability limit state, nef.ser, is used in order to take into account possible 188
group effects and its value is discussed in Section 3.2.
189
The withdrawal capacity, Pu.w, is given by Eq. (4):
190
sin( ) tanh[ (1 )] cos( )
=
u.w ef
n u n u n u
ef u.w.singleef w n n
P m ω λ ω - λ m ω λ
n + n P
π d l f ω m ω
(4)where fw is the withdrawal strength and 𝑚 = √𝛤𝑓⁄𝛤𝑒 is a parameter which expresses the brittleness of the shear 191
zone. Note that nef has been used in Eq. (4) in order to take into account possible group effect on capacity of 192
multiple rods, and its value is discussed in Section 3.1.
193
The parameter ωn is defined as:
194
= 2
n e n ef
ω π d Γ β l (5)
where βn is given by:
195
= 1 +
n
s s w w.α
β n
A E A E
(6)
9
Here Es and Ew.α are the moduli of elasticity of steel and wood (as function of α), respectively. Τhe core cross- 196
sectional area of the rod is As = π∙dc2/4 and Aw is the area of wood subjected to axial stress. Ew.α may be estimated 197
by the Hankinson formula and Aw by an effective area, confer (Stamatopoulos and Malo 2016).
198
The effective length lef and the parameters Γe (in N/mm3), fw (in MPa) and m and are given by Equations (7)-(10) 199
(Stamatopoulos 2016):
200
= - 0.5
l
efl d
(7)= 9.65
1.5 sin cos
e.α 2.2 2.2
Γ α + α
(8)= 4.70
0.95 sin cos
w.α 2.2 2.2
f α + α
(9)0.332
=1.73 sin + cos mα
α α (10)
Note that, in principle, the parameters Γe and Γf for withdrawal of multiple rods are different from the single rod 201
case due to stresses’ interaction. However, for rods installed ‘in parallel’ the difference is assumed to be small.
202
For rods installed in small angles to the grain there is a high shear stress concentration in the vicinity of the 203
interface (i.e. the magnitude of shear stresses is much higher near the interface). For rods installed with greater 204
angles to the grain, the shear stress distributes mainly along the strong shear plane. Therefore, Eq. (8) is assumed 205
to be a good approximation. Possible group effects may indirectly be taken into account by the use of nef.ser in Eq.
206
(3). Note that such an approach will provide the same estimation for a given angle, regardless of the 207
configuration of rods with respect to edge distances and spacings. The parameter λu is a dimensionless length 208
parameter which expresses the percentage of the embedment length at failure in which fracture behaviour takes 209
place and it can be determined by the diagram given in Fig. 5.
210
211
212
Fig. 5 Diagram for the determination of λu (Stamatopoulos and Malo 2015b) 213
10 2.3 NUMERICAL
214
Finite element simulations were performed to estimate the withdrawal stiffness as well as the stress and 215
displacement distributions in all specimens. Abaqus software (Abaqus analysis user's guide, Version 6.13 2013) 216
was used for the finite element simulations. The finite element model assembly is visualized in Fig. 6a. It 217
consists of a rectangular box-type timber part in surface contact with the embedded threaded rods. The threads 218
and the core of the threaded rod were meshed independently in separate sub-parts, which were jointed using a tie 219
constrain as shown in Fig. 6b. Similarly, the timber part was created by tying two independently meshed sub- 220
parts; a sub-part with the respective female thread geometry of the rod and an exterior timber sub-part. These 221
two parts were tied in the interface between the timber part and the outer thread surface of the rod; confer the 222
detail shown in Fig. 6c. The timber part was more densely meshed in the vicinity of the interface with the 223
threaded rod. The mesh size gradually increased with increasing distance from the interface. Three dimensional, 224
8-node, linear brick elements were used to mesh all parts. Each threaded rod was loaded by a unit vertical pull- 225
out force, P = 1 kN. Lateral displacements of the rods at the loading point were restrained.
226
Wood is an anisotropic material, which can be approximated as orthotropic with three distinct material 227
orientations; the longitudinal (parallel to the grain), the radial and the tangential (with respect to the annular 228
rings). The subscripts L, R and T are used to indicate these material directions. However, in these simulations 229
wood was modelled as transversely isotropic (in Cartesian coordinates), assuming equal properties in the radial 230
and the tangential directions. The material properties given in Table 3 were used for the simulations. Due to an 231
incomplete set of material properties provided by the manufacturer, the lacking properties were taken from a 232
study on mechanical properties of Norwegian spruce (Dahl 2009). The steel of the rods was modelled as 233
isotropic with modulus of elasticity equal to Es= 210 GPa and Poisson’s ratio equal to 0.30. Both steel and wood 234
were modelled as linear-elastic. The contact interaction between the wood and the rod was modelled with hard 235
contact normally to the surface and frictional behaviour tangentially. For the normal contact, the augmented 236
Lagrange method was used as constraint enforcement method. The friction coefficient for the wood-steel surface 237
was set equal to μ= 0.20. This decision was supported by the study of (Koubek and Dedicova 2014) who 238
investigated the friction coefficient of wood products (laminated veneer lumber and pine wood) as function - 239
among others - of the angle to the grain and contact pressure. This study showed that for normal moisture 240
content and pressure parallel to the grain, the friction coefficient is approximately equal to 0.25 and 0.20 for low 241
and high values of the contact pressure, respectively. For other angles, the friction coefficient was found to be 242
smaller. Depending on the specimen the friction coefficient was in the range of 0.15-0.30 for low contact 243
pressure and in the range of 0.12-0.22 for higher contact pressure. The numerical results for varying friction 244
coefficient in these ranges are very similar and therefore a constant value of μ= 0.20 was assumed to be a 245
reasonable input value for the Finite Element simulations.
246
11 247
Fig. 6 Numerical simulation: (a) model assembly (b) finite element model of the threaded rod, and (c) detail of 248
the finite element model of the timber part 249
250
Table 3: Material properties for numerical simulation 251
Material property Symbol Value Input for Simulation (R≡T)
Mean density (kg/m3) ρm 470 a 470
Moduli of Elasticity (MPa)
EL ≡ Ew.0 13000 a 13000
ER = ET ≡ Ew.90 410 a 410
Shear Moduli (MPa) GLR = GLT 760 a 760
GRT 30.7 b 30
Poisson ratios
νLR 0.501 b
0.60 νLT 0.695 b
νTR 0.315 b
0.60 νRT 0.835 b
a Values provided by the manufacturer
b Values by (Dahl 2009)
252 253
3 RESULTS AND DISCUSSION 254
3.1 WITHDRAWAL CAPACITY 255
All specimens failed due to withdrawal of the rods. Typical failure modes for each specimen series are depicted 256
in Fig. 7. For specimens with α ≠ 90° a splitting crack formed along the grain, confer the photos (a)-(f) in Fig. 7.
257
12
However, this crack was bridged by the rods and did not appear to have a strong influence on the structural 258
behaviour.
259 260
261
Fig. 7 Typical failure modes of series (a) S15-A, (b) S15-B, (c) S30-A, (d) S30-B, (e) S60-A, (f) S60-B, (g) S90- 262
A and (h) S90-B 263
264
The experimental results for the withdrawal capacity are summarized in Table 4. The withdrawal capacity for 265
each specimen and the mean capacity for each configuration and angle are provided. Treating all experimental 266
results for the withdrawal capacity as one sample, a coefficient of variation (abbr. CoV) equal to 5.3% is 267
obtained. This value of CoV is quite similar to the case of single rods if small values of α-which are inherently 268
variable-are excluded (for example by use of the experimental results of the reference single rods study 269
(Stamatopoulos and Malo 2015b) we obtain CoV= 8.8% if results for α= 0° are excluded and CoV= 6.2% if 270
results for α= 0°, 10° are excluded for rods with l= 450 mm). Thus the capacity is quite reliable and its variability 271
is similar to the reliability in the case of single rods and therefore approximate conclusions can be obtained 272
despite the small sample size.
273
For specimens with α= 60°, 90° the mean withdrawal capacity of specimens with the configuration A was 274
greater by 0.9% and 0.4% respectively, compared to specimens with the configuration B. For specimens with α=
275
30°, the experimentally recorded mean withdrawal capacity was greater for configuration B by 4.5% compared 276
to configuration A. One recording was lost for a specimen with α= 15° and rods installed in the configuration B, 277
13
and thus a reasonable comparison between the configurations for α= 15° is not possible. As seen by the results in 278
Table 4 the withdrawal capacity is characterized by very small variability.
279
Characteristic values for the withdrawal capacity were also calculated by considering all results per angle as one 280
sample, i.e. without distinguishing between different configurations. CoV was very small for all samples (7% for 281
α= 15°, 3% for α= 30 and 60°, and 1% for α= 90°). The characteristic values were determined in accordance 282
with EN 14358 (CEN 2006) and they are also provided in Table 4. EN 14358 (CEN 2006) assumes that the test 283
values are log-normally distributed. Despite the small size of each sample the calculated characteristic 284
withdrawal capacities are assumed to be a reasonable approximation, due to the use of a minimum value of 0.05 285
for the standard deviation of the natural logarithms of test values, according to EN 14358 (CEN 2006) in cases 286
where CoV is smaller than 5%.
287
The effective numbers of rods nef (both for mean and characteristic values) are also given in Table 4. They were 288
determined by use of results from the reference single rod experimental investigation of specimens with the same 289
threaded rods and the same glulam strength class as in the present investigation (Stamatopoulos and Malo 2015a;
290
2015b). The individual and mean values of nef are plotted as function of α in Fig. 8. Based on the obtained mean- 291
level experimental results the following approximating expression was derived for nef: 292
0.9
°) 1.75 0.116 ( / 60
= =1.8
<
6
60°
60°
6
ef
, α , α
+ α
n n
(11)
The analytical estimations (also provided in Table 4) were made according to Eq. (4), using the effective number 293
of rods provided by Eq. (11). The experimental results together with analytical estimation are plotted as function 294
of α in Fig. 9. As shown in these figures they are in good agreement with the analytical estimations being 295
slightly conservative.
296
Table 4: Results-withdrawal capacity Pu.w (kN) 297
Specimen series
Experimental Analytical
mean a Test 1 Test 2 Mean (nef b) Characteristic (nef c)
S15-A-(1-2) 247.5 223.5 235.5 (1.72)
191.5 (1.79) 228.6
S15-B-(1-2) 258.9 (-) d 258.9 (1.89)
S30-A-(1-2) 250.1 260.3 255.2 (1.77)
226.7 (1.96) 243.4
S30-B-(1-2) 265.6 267.8 266.7 (1.84)
S60-A-(1-2) 277.8 271.4 274.6 (1.94)
237.5 (1.90) 257.9
S60-B-(1-2) 283.0 261.5 272.2 (1.92)
S90-A-(1-2) 259.0 263.7 261.4 (1.88)
226.7 (1.86) 243.7
S90-B-(1-2) 261.3 259.5 260.4 (1.87)
a Values for analytical approach (Stamatopoulos and Malo 2016):
Ew.0 =13000 MPa, Ew.90 =410 MPa, Ew.α = Ew.0· Ew.90 / (Ew.0·sin2α + Ew.90·cos2α), Es = 210000 MPa Aw ≡ Aw.eff = 2·140·(180 + 450/6) = 71400 mm2 , As = π·dc2 /4 = 176.6 mm2
b Mean experimentally recorded capacities of specimens with single rods (Stamatopoulos and Malo 2015b):
Pu.w.15 = 136.7 kN (mean of Pu.w.10 and Pu.w.20), Pu.w.30 = 144.6 kN, Pu.w.60 = 141.7 kN, Pu.w.90 = 139.2 kN
c Characteristic experimentally recorded capacities of specimens with single rods (Stamatopoulos and Malo 2015a):
Pu.w.15.k = 106.7 kN (mean of Pu.w.10.k and Pu.w.20.k), Pu.w.30.k = 115.5 kN, Pu.w.60.k = 125.2 kN, Pu.w.90.k = 121.9 kN
d Recording was lost
14 298
Fig. 8 Experimentally determined values of nef as function of α 299
300
301
Fig. 9 Withdrawal capacity as function of α 302
303 304
3.2 WITHDRAWAL STIFFNESS 305
The experimental results for the withdrawal stiffness together with the analytical and the numerical estimations 306
are summarized in Table 5 and plotted as function of α in Fig. 10. The specimens exhibited very high withdrawal 307
15
stiffness especially for small angles. Note that only a small number of tests have been performed and hence the 308
measured values for stiffness may not be representative in general. Analytical estimations were made according 309
to Eq. (3) assuming two different effective number of rods; the actual number of rods (nef.ser= 2) and the effective 310
number of rods given by Eq. (2). In general, nef.ser= 2 provided a better analytical estimation than Eq. (2). The 311
agreement between experimental results and the analytical and numerical estimations is very good. According to 312
the experimental results, the difference between the values for configurations A and B was relatively small 313
(configuration B was stiffer by 7.1%, 14.6% and 1.4% for α= 15°, 30° and 90° respectively while configuration 314
A was stiffer by 5.3% for α= 60°) . The distributions of stresses and displacements along the rod were quantified 315
by the numerical simulations. These distributions were essentially the same as the distributions for the single rod 316
case (Stamatopoulos and Malo 2016) and therefore they are not presented in the present paper.
317 318
Table 5: Results-withdrawal stiffness Kw (kN/mm) 319
Specimen series
Experimental Analytical a Numerical Test 1 Test 2 Mean nef.ser = 2 nef.ser = 20.8
S15-A-(1-2) 299.7 220.6 260.2
258.5 225.0 273.7
S15-B-(1-2) 318.3 237.8 278.0 281.1
S30-A-(1-2) 170.4 226.3 198.3
219.5 191.1 233.6
S30-B-(1-2) 184.5 270.0 227.3 239.9
S60-A-(1-2) 118.1 149.3 133.7
151.8 132.5 151.6
S60-B-(1-2) 131.3 122.5 126.9 157.2
S90-A-(1-2) 144.8 108.8 126.8
129.2 112.5 127.4
S90-B-(1-2) 139.1 118.0 128.6 132.5
a Values for analytical approach: same as in Table 4 320
321
Fig. 10 Withdrawal stiffness as function of α 322
16 4 CONCLUSIONS
323
The withdrawal of pairs of axially loaded threaded rods screwed into glued-laminated timber elements was 324
studied. The rods were installed in ‘parallel’, i.e. in a row perpendicular to the plane of the grain. Specimens with 325
two configurations with respect to the edge distances and spacings were tested; one with small spacing between 326
the rods (configuration A) and one with small edge distances (configuration B). The edge distances and spacings 327
were smaller than the minimum values required by EC5 (CEN 2004). The outer thread diameter and the 328
embedment length of the threaded rods were d= 20 mm and l= 450 mm, respectively. Specimens with 4 different 329
rod-to-grain angles were tested (α= 15, 30, 60, 90°). Analytical and numerical estimations were compared to the 330
experimental results. The following main conclusions are drawn:
331
Interaction effects (group effect) of the rods were approximated by use of an effective number of 332
rods nef. 333
The values of nef, were evaluated on the basis of experimental results and a simple approximating 334
expression for its determination was derived. Despite very small edge distances and spacings the 335
mean values of nef per configuration and angle were in the range 1.72-1.94.
336
Based on the obtained experimental results, the difference between the results for configurations A 337
and B was-in general-small.
338
The withdrawal capacity and stiffness can be estimated by an analytical model which is based on 339
Volkersen model with an idealized bi-linear constitutive relationship.
340
The withdrawal stiffness can be estimated with sufficient accuracy by finite element simulation.
341 342
ACKNOWLEDGEMENTS 343
The support by The Research Council of Norway (208052) and The Association of Norwegian Glulam 344
Producers, Skogtiltaksfondet and the Norwegian Public Road Administration is gratefully acknowledged.
345 346
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